Problem: What is the value of $\dfrac{d}{dx}\left(x^{^{\scriptsize\dfrac{3}{2}}}\right)$ at $x=9$ ?
Answer: Let's first find the expression for $\dfrac{d}{dx}\left(x^{^{\frac{3}{2}}}\right)$ and then evaluate it at $x=9$. The derivative can be found using the power rule : $\dfrac{d}{dx}(x^n)=n\cdot x^{n-1}$ (Remember that this applies even when $n$ is a fraction.) $\begin{aligned} &\phantom{=}\dfrac{d}{dx}\left(x^{^{\frac{3}{2}}}\right) \\\\ &=\dfrac{3}{2}x^{^{\frac{3}{2}-1}} \gray{\text{The power rule}} \\\\ &=\dfrac32x^{^{\frac{1}{2}}} \end{aligned}$ So we found that $\dfrac{d}{dx}\left(x^{^{\frac{3}{2}}}\right)=\dfrac32x^{^{\frac{1}{2}}}$, which can also be written as $1.5{\sqrt{x}}$. Now let's plug ${x=9}$ : $\begin{aligned} 1.5\sqrt{{9}}&=1.5\cdot 3 \\\\ &=4.5 \end{aligned}$ In conclusion, the value of $\dfrac{d}{dx}\left(x^{^{\frac{3}{2}}}\right)$ at $x=9$ is $4.5$.